Correlated/dependent random variables This may be a very basic question or a very difficult one, I am not sure.
What does it mean if two random variables are perfectly correlated?
What does it mean if two random variables are perfectly dependent (whatever that means)?
When two random variables are independent joint pdf can be factorized into marginal pdfs. But if they are say perfectly correlated, what does that say? Does it mean they come from  distributions with same functional forms but with different variances?
What does it mean to be perfectly dependent? Conditional distribution collapses to a dirac delta function?
Thanks
 A: I think one could interpret "two random variables are perfectly correlated" as "the correlation of the two random variables is $1$ (or maybe $\pm 1$)", and
interpret "the two random variables are perfectly dependent" as "two random variables are comonotonic (or maybe comonotonic or countermonotonic)"
Perfect linear dependence
Let $X$ and $Y$ be two random variables. If $|\mbox{corr}(X,Y)| = 1$, then $Y = \alpha + \beta X$ almost surely for some $\alpha \in \mathbb{R}$ and $\beta \neq 0$, with $\beta > 0$ for positive linear dependence and  $\beta < 0$ for negative linear dependence.
It can be shown that $\mbox{corr}(X,Y) = 1$ if and only if $X$ and $Y$ are of the same type, that is there exists constants $a\in \mathbb{R}$ and $b > 0$ such that $Y \stackrel{\mathrm{d}}{=} a + bY$, where $\stackrel{\mathrm{d}}{=}$ denote equallity in distribution, meaning that $X$ and $Y$ have the same distribution up to a location and scale transformation.
Perfect positive dependence
The random variables $X$ and $Y$ are comonotonic if they admit as copula the Fréchet upper bound, that is the copula associated to $X$ and $Y$ is $C(u,v) = \min(u,v)$, which is the strongest type of "positive" dependence.
It can be shown that $X$ and $Y$ are comonotonic if and only if
$$
(X,Y) \stackrel{\mathrm{d}}{=} (h_1(Z), h_2(Z))
$$
for some random variable $Z$ and increasing functions $h_1$ and $h_2$. So, comonotonic random variables are only functions of a single random variable; the conditional density of $Y$ given $X = x$ is a Dirac delta function.
The joint distribution of $X$ and $Y$ is
$$
F(x,y) = C(F_X(x), F_Y(y)) = \min(F_X(x), F_Y(y)) ,
$$
where $F_X$ and $F_Y$ denote the marginal distributions of $X$ and $Y$.
